Question

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Prob. 1 Plot 3e-10.25on a circle in the complex plane at t-k, fork-0-5. What is its direction of rotation, clockwise or counterclockwise? What are its period P (sec) and its (fundamental) frequency in rad/s and in Hz? Prob. 2 Is the signal X(t)- 5+ 2sin4.2 + 4cos10.5t periodic? If it is not explain. If it is, what is its fundamental period P, and fundamental frequency wo (rad/s) and (Hz) ? (Hint: You can use Octave to find prime factors of an integer. For example, if you type: factor(123) you'll sce: ans - 3 41. since 123-341.)

Answer 1

Problem 1. Given equation is of circle whose magnitude is 3 and phase is -0.25$\pi$t. Since it is negative so its direction is clockwise. Its fundamental frequency is 0.25$\pi$rad/sec and in Hz 0.125Hz and Time period P is 2$\pi$/0.25 which is 8sec.

substituting t with k and taking values of k from 0 to 5 , magnitude remains same but phases change as from 0, -$\pi$/4, -$\pi$/2, -3$\pi$/4, -$\pi$,-5$\pi$/4

problem 2. Given signal x(t) comprises of three signals in which one signal is a constant and the other two are normal periodic signals because ratio of their periods is a rational number. Although constant signals are always periodic, their period is undefined. Hence given signal is periodic but its period will remain undefined because of constant signal, we cannot take ratio of its period with periods of other signals comprising x(t).

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